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Simplify sin (𝜃 − 90°).

Simplify the sin of 𝜃 minus 90 degrees.

There are several ways of simplifying this expression. We will begin by using the cofunction and odd identities. One of the cofunction identities states that the sin of 90 degrees minus 𝜃 is equal to cos 𝜃. Factoring negative one out of the expression inside our parentheses gives us sin of negative 90 degrees minus 𝜃. We recall that the sine function is odd such that the sin of negative 𝛼 is equal to negative sin 𝛼. This means that we can rewrite our expression as negative sin of 90 degrees minus 𝜃. Using the cofunction identity, this is therefore equal to negative cos 𝜃. The sin of 𝜃 minus 90 degrees is equal to negative cos 𝜃.

An alternative method here would have been to have used the difference formulae for the sine function. This states that sin of 𝛼 minus 𝛽 is equal to sin 𝛼 cos 𝛽 minus cos 𝛼 sin 𝛽. Substituting 𝛼 as 𝜃 and 𝛽 as 90 degrees, we have sin 𝜃 multiplied by cos of 90 degrees minus cos 𝜃 multiplied by sin of 90 degrees. The cos of 90 degrees is zero, and the sin of 90 degrees is one. This gives us sin 𝜃 multiplied by zero minus cos 𝜃 multiplied by one, which is equal to negative cos 𝜃. This confirms the answer we found using the cofunction and odd identities.

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